Functional equations and gamma factors of local zeta functions for the metaplectic cover of SL2

Abstract

We introduce a local zeta-function for an irreducible admissible supercuspidal representation π of the metaplectic double cover of SL2 over a non-archimedean local field of characteristic zero. We prove a functional equation of the local zeta-functions showing that the gamma factor is given by a Mellin type transform of the Bessel function of π. We obtain an expression of the gamma factor, which shows its entireness on C. Moreover, we show that, through the local theta-correspondence, the local zeta-function on the covering group is essentially identified with the local zeta-integral for spherical functions on PGL2≅SO3 associated with the prehomogenous vector space of symmetric matrices of degree 2.